What is the integral of 2sqrt(x)(10x+3) ?

The integral of 2sqrt(x)(10x+3) is 2(4x^{5/2}+2x^{3/2})+C

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integral of 2sqrt(x)(10x+3)

\int 2 x (10 x + 3) d x

2 (4 x^{\frac{5}{2}} + 2 x^{\frac{3}{2}}) + C

Solution steps

\int 2 x (10 x + 3) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int x (10 x + 3) d x

Expand

x (10 x + 3) : 10 x^{\frac{3}{2}} + 3 x

= 2 \cdot \int 10 x^{\frac{3}{2}} + 3 x d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int 10 x^{\frac{3}{2}} d x + \int 3 x d x)

\int 10 x^{\frac{3}{2}} d x = 4 x^{\frac{5}{2}}

\int 3 x d x = 2 x^{\frac{3}{2}}

= 2 (4 x^{\frac{5}{2}} + 2 x^{\frac{3}{2}})

Add a constant to the solution

= 2 (4 x^{\frac{5}{2}} + 2 x^{\frac{3}{2}}) + C

What is the integral of 2sqrt(x)(10x+3) ?
The integral of 2sqrt(x)(10x+3) is 2(4x^{5/2}+2x^{3/2})+C